If the goal is to achieve more effective democracy, it is not easy to make a strong case for most of these alternatives. Random assignment to an arbitrary group would be an approach. So would whole constituency "single districting." All have obvious flaws; some systems are hard to game but difficult to operate with, others are terribly easy to game.
What is needed, if the goal is to achieve democracy where the individual voter can most effectively empower representatives they believe will best promote their individual priorities in the legislature, is a safety valve as it were whereby misapportionment does not neutralize blocs of voters.
I'd propose a system in which a breakdown of the electorate by some system--it almost does not matter which--divides the voters up into half as many bins as there are seats, rounding down if there are uneven numbers of seats in the body to be elected. Allow very wide open nomination; there can be restrictions on who is conveniently listed on the ballot but must then be provision for a clear "write-in" system. Every party or interest group that can qualify for having a list of recognized candidates is assigned a random identifier code so that relatively obscure groups who can't prove popularity before a given general election have an unambiguous way of identifying themselves, that their supporters in any district or other sort of "bin" they are in can identify who they are voting for. For they are not restricted to voting for someone in their own "bin!"
Then everyone is entitled to vote for two individuals. They are not obligated to identify two individuals; they can vote for a party or listed interest group if they like; there have to be rules then to identify which individual in that list gets the vote. This might be left up to the party or interest group administrators and can be as arbitrary as they like--all generic votes go to the leading individuals, or are randomly assigned to their lists, or assigned to someone in the middle according to an algorithm for maximizing seated members--the important rule is that their system for doing it be published and strictly adhered to. But "the party leader assigns them with full discretion" would be a valid published rule. Voters get to decide who they trust and who they don't and are assumed to know what they are doing; with such broad freedom of choice they can be as picky as they like.
Then the candidates are assigned proportionately, by means of listing the factions in order of how many votes they got, and modifying the list with divisors as the seats are assigned to each, much as the number of Representatives each US state is apportioned by population is managed. Within each faction, the individual candidate as yet unseated who has the highest number of votes takes the next seat their faction gets, and either the higher divisor their faction has shoves them some distance down the list opening the next seat to another faction, or else the next highest vote getting candidate gets the next seat, and so on one way or the other. The results of such an allocation are very similar to what happens if we simply assign each faction a number of seats in proportion to their faction's proportion of votes, but the question of how to appropriate fractional results is resolved by the rotation process.
With such a system, it does not matter much how the apportionment of the half of seats bins has gone. Suppose we had a city where it was decided that alphabetical apportionment would work best for instance. A glance at a telephone directory indicates that surnames are not evenly distributed across the alphabet; if it were desired to seat a 51 member City Council and there was none of this proportionality stuff, just winner-take-all for 51 separate "bins" of voters, it would be necessary to meticulously examine the list of registered voters, determine how far down it from the letter A (assuming a Latin alphabet using electorate here) one runs before reaching 1.96 percent of the electorate, note which pair of names this division falls between, and so on for 50 more fussy distinctions. Entire strings of letters might be included in one bin while other letters are subdivided into two, or even three. The boundaries will rarely in fact coincide with the end of one letter and the beginning of another. What happens if a division falls within a very common name like Johnson or Smith? Do individual Johnsons or Smiths get assigned randomly according to a supplementary rule, or do we lump them all into whichever category happens to contain more of them--thereby misapportioning, as the category with all the Smiths or Johnsons at one end or the other is now overly large, while the one preceding or following is now scanted of voters--hence its voters are now more powerful than they should be?
However painstaking a job we do in locating the fair break points between each group (using first name spellings, as far as is necessary to go, to subdivide the Johnsons for instance, to get even groups) as voters die or move away, and others move in or come of voting age, the balances will be upset and by election day some groups will be larger with their individual voters suffering diluted votes while others will be smaller thus enjoy concentrated power. It is the same with any other system of course. And she of those proposed would inherently tend to institutionalize such biases.
But now look what happens if instead we have 51 seats but simply declare that everyone votes in a "bin" determined simply by which of the 26 letters their surname begins with? Now the bins are wildly malapportioned of course! Some letters are more common as heads of surnames than others and these will be underrepresented, while those with surnames starting with rarer letters will be absurdly overempowered. But there are not 26 seats to fill, there are 51, and anyone can vote for any two in any group! Their ballots, prepared for them, will only list the leading candidates in their letter category. But they remain free to learn the names and faction code of whomever they like. The assignment of the population to categories is just an arbitrary organizational tool. Suppose for the moment that voters all decide to stick to voting for those who show on their ballots. It might be the case that the people whose name starts with "M" outnumber the people whose name starts with "Y" six to one. But even if, as in typical American elections, most everyone votes for just one of three parties, the M voters will count six times as much as the Y voters because there are more of them. The people in the large categories will have power proportional to their numbers, the small categories will be outnumbered. First the large categories will largely determine which factions come first in the list, and secondly their candidates, having received far larger numbers of votes, will head the lists within each faction of who gets seated first. The small category people are rather unlikely to see anyone they voted for seated in the council at all, but on the other hand they do have their due influence in terms of which factions are where on the list of total vote-getters. Furthermore once we allow people to vote for whichever candidates they like via write-ins, it becomes possible for an individual candidate who happens to come from a small category to nevertheless get votes from outside their category, and thus come in high, perhaps at the top of, their faction's list.
If instead of alphabetical assignment we had geographical districts, it should be clear how it is that many traditional manipulations of these districts are almost automatically neutralized by choosing the seats proportionally across the electorate in this way. The grossest way to game districts is of course to malapportion them, to define very large districts that contain more than their share of voters and thus define others that contain far fewer--this sort of thing was very common in US history and typically resulted in empowering the rural countryside voters at the expense of big cities. It could of course go the other way in theory and probably has in practice, somewhere or other. Either way is grossly unfair in a district system, yet once established it can very strongly perpetuate itself without some kind of forceful intervention.
By assigning seats proportionally instead, even gross disparities in district sizes would matter a lot less, and the dynamic of unfairness that remains is reversed--now it is the big districts that get first pick of seats. However--individual candidates getting much larger numbers of votes because they come from big districts don't get more power in the legislature than one seat; to express the power of their factions, people from their factions who got votes in the underpopulated districts will be seated on their coattails as it were, so the underpopulated regions do have some leverage. The prospects of the political process correcting the misapportionment are much better.
Another way is of course to gerrymander; to create districts that are indeed more or less the right balance of total population, but to attempt to dilute the power of particular groups, either by deliberately splitting them up to dilute them in districts where others are the majority, or by concentrating them all in as few districts as possible to limit their numbers below their fair share of the population in the legislature. Again, the automatic linkage of all votes into one big pot goes very far toward automatically foiling such an agenda. If the group discriminated against has a strong consciousness of their targeted status, they are of course quite likely to band together, and thus even if none of them can win a district, their combined votes from the many districts they are in can elect a proportional number of legislators, who can then protest the gerrymandering in that body. And if they can't get a majority to support eliminating the deliberately unfair districting, they can anyway live with it and with organization get their fair share of representation anyway.
I came to this thread keenly interested in alternative systems, but I think all the suggestions offered thus far highlight why it is that geography is usually deemed the best method. If the US Congress were elected proportionally by at large votes, for instance, we might have no effective means for addressing strictly regional issues, once the regions in question are too small to command large blocs of voters. With the kind of geographical dispersion we have, even today with speed of light telecommunications and the ability to travel from one end of the nation and back again within a single day (even leaving some time at both ends to get some business done) I'd think we'd be badly served to get rid of the regional principle completely. But with an integrating proportional system such as I describe that begins with a territorial basis, the drawbacks of geographical systems are largely removed, but their benefits remain in place. If in fact the preoccupying issues that trouble most voters in a given district are local in nature, such a system allows them to vote with a regional agenda in mind and have effective power on that basis. If on the other hand even only a small minority in these regions disagree, and would rather support Representatives who have a broader agenda, on other issues entirely, they can effectively do that.
Most of these benefits come from the system being proportional as such. I am not sure though that the sort of PR I have described here is in practice or widely understood by some standard name.
I've particularly been at pains to emphasize alternative modes of nomination for voter consideration to party processes. By and large I think party affiliation is basically a good thing, but it is very important to have some means of going outside of regular party processes, so that disgruntled voters always have somewhere to go. Otherwise oligarchy can easily game the system by controlling party nomination processes. So I would urge incorporation of alternative nomination processes that can get alternative accountable lists of candidates before the voters (even if obscurely) in the general election.
This general framework I have been trying to describe could accommodate itself to any of the alternatives to geographical representation offered. Consider the option of dividing the electorate up by nationalities, ethnicities, or religion for instance. Clearly such a process is open to abuse on a massive scale, if what is created is winner take all seats! To the invidiousness of gerrymandering and misapportionment we add the narrative of essentialist divisions. An ethnic based process has actually looked to me like possible creative and workable solutions, perhaps--in the Austro-Hungarian empire for instance. But what of borderline cases? Who gets to decide if I am a Jew, a Bosnian, or an Austrian? Do I get any say in the matter?
Again if the categories are only a framework to hang a free proportional election process on, in which I get to determine the ultimate outcome as far as my vote goes, it matters a lot less. As a Catholic Austrian I can vote for a Hungarian Jew or a Bosnian Muslim if that seems right to me, or in voting for some "fellow" Catholic Austrian I can support a faction unpopular in my category but with strong alliance to a popular majority in some other category, thereby supporting the latter indirectly. Vice versa if the homogenizing agenda an enlightened Hapsburg monarchy would presumably be working toward were to succeed, people in all these diverse categories can converge without impediment on a common moderate party thereby softening and perhaps practically eliminating the invidiousness of subdividing the populace into these groups.
Suppose we were to adopt Marius's suggestion that people self-identify by ideology? Again if the formal process allows individual voters to cast their votes very freely, outside their category if they so choose, and there is an open nomination process alternative to internal party processes so that no group of voters can be captured against their will, the obvious drawbacks of a system demanding people publicly declare their affiliation to be enfranchised can be neatly sidestepped. I can register as a Republican in Arizona (in a system where that identity is very tightly defined unlike our OTL one where it is in principle very broad) because I know my employers and other powers that be would frown on me if I did otherwise--but then anonymously vote for some radical left candidate in Minnesota.
And so on. I do think that with such a system in place the most sensible mode of "binning" the populace for the substrate of the process would be geographical. It might be otherwise in specific contexts. A Labor party for instance might be best organized by subdividing first along trade lines, only devolving to regional within these categories.